A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

Nonlinear phenomena exist in all areas of science and engineering, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, and chemical physics. It is well known that many nonlinear partial differential equations (NLPDEs) are widely used to describe these complex physical phenomena. The exact solution of a differential equation gives information about the construction of complex physical phenomena. Therefore, seeking exact solutions of NLPDEs has long been one of the central themes of perpetual interest in mathematics and physics. With the development of symbolic computation packages, like Maple and Mathematica, many powerful methods for finding exact solutions have been proposed, such as the homogeneous balance method [

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations. It has been developed by Kudryashov [

In this paper, we first apply the simplest equation method and the modified simplest equation method to derive the exact solutions of the elliptic-like equation, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained.

Suppose that we have a nonlinear partial differential equation (PDE) for

By taking

Suppose the solution

The Bernoulli equation we consider in this paper is

For the Riccati equation

Substituting (

Assuming that the constants

In the modified version, one makes an ansatz for the solution

Substitute (

Now, let us choose the following elliptic-like equation

Considering the homogeneous balance between

Substituting (

Therefore, using solutions (

Suppose the solutions of (

Substituting (

Therefore, using solutions (

Suppose the solution of (

Substituting the values of

Solving (

And solving (

When

When

Upon integration, we obtain

From (

We can arbitrarily choose the parameters

Again setting

Using hyperbolic function identities, from (

Using

Integrating (

As (

From (

Based on the conclusion just mentioned, we only solve (

Equation (

Then, solutions of (

Consider

We assume that

Substituting (

Integrating (

Substituting (

Equation (

Then, solutions of the Klein-Gordon-Zakharov (KGZ) system are defined as follows:

We consider a class of NLPDEs with constant coefficients [

If one takes

Since

If we set

Integrating (

Equation (

Then, solutions of (

We may obtain from (

This equation coincides also with (

We may obtain from (

This equation coincides also with (

The simplest equation method is a very powerful mathematical technique for finding exact solutions of nonlinear ordinary differential equations, and the elliptic-like equation is one of the most important auxiliary equations because many nonlinear evolution equations, such as the perturbed nonlinear Schrödinger's equation, the Klein-Gordon-Zakharov system, the generalized Davey-Stewartson equations, the Davey-Stewartson equations, the generalized Zakharov equations, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system, and the generalized ZK-BBM equation, can be converted to this equation using the travelling wave reduction.

In this paper, we apply the simplest equation method and the modified simplest equation method to derive the exact solutions of the elliptic-like equation. The exact solutions of the perturbed nonlinear Schrödinger's equation, the Klein-Gordon-Zakharov system, the generalized Davey-Stewartson equations, the Davey-Stewartson equations, and the generalized Zakharov equations are derived. Comparing the currently proposed method with other methods, such as the

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (11161020; 11361023), the Natural Science Foundation of Yunnan Province (2011FZ193; 2013FZ117), and the Natural Science Foundation of Education Committee of Yunnan Province (2012Y452; 2013C079).